Hanoi attractors and the Sierpiński gasket

The famous game Towers of Hanoi is related with a family of so-called Hanoi-graphs. We regard these non-self-similar graphs as geometrical objects and obtain a sequence of fractals (HGα)α converging to the Sierpinski gasket which is one of the best studied fractals. It is shown that this convergence holds not only with respect to the Hausdorff distance, but that also Hausdorff dimension does converge. Moreover, it is shown that each of the approximating sets has non-trivial Hausdorff measure.